Evaluation of one exotic Furdui type series

This is a closed form evaluation of one interesting alternating series whose value is a combination of three mathematical constants, Pi, log 2, and Zeta (3).Comment: Updated version of the paper in the Far East Journal of Mathematical Science

A special constant and series with zeta values and harmonic numbers

In this paper we demonstrate the importance of a mathematical constant which is the value of several interesting numerical series involving harmonic numbers, zeta values, and logarithms. We also evaluate in closed form a number of numerical and power series.Comment: 19 page

Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials

This is a short survey of a class of functions introduces by Tom Apostol. The survey is focused on their relation to Eulerian polynomials, derivative polynomials, and also on some integral representations

Derivative Polynomials for tanh, tan, sech and sec in Explicit Form

The derivative polynomials for the hyperbolic and trigonometric tangent, cotangent and secant are found in explicit form, where the coefficients are given in terms of Stirling numbers of the second kind. As application, some integrals are evaluated and the reflection formula for the polygamma function is written in explicit form.Comment: A similar version in The Fibonacci Quarterly, 200

Evaluation of some simple Euler-type series

Five series are evaluated in terms of zeta values. Three of the series involve harmonic numbers and one involves Stirling numbers of the first kind. The evaluation of these series is reduced to the evaluation of certain integrals, including the moments of the polylogarithm

On a series of Furdui and Qin and some related integrals

In this note we present the solution of Problem H-691 (The Fibonacci Quarterly, 50 (1) 2012) with some corrections and more details. The solution involves three nontrivial integrals whose evaluations are given here

Eratosthenes and Pliny, Greek geometry and Roman follies

Supportive attitudes can bring to a blossoming science, while neglect can quickly make science absent from everyday life and provide a very primitive view of the world. We compare one important Greek achievement, the computation of the Earth meridian by Eratosthenes, to its later interpretation by the Roman historian of science Pliny.Comment: 9 page

Binomial transform and the backward difference

We prove an important property of the binomial transform: it converts multiplication by the discrete variable into a certain difference operator. We also consider the case of dividing by the discrete variable. The properties presented here are used to compute various binomial transform formulas involving harmonic numbers, skew-harmonic numbers, Fibonacci numbers, and Stirling numbers of the second kind. Several new identities are proved and some known results are given new short proofs.Comment: The paper is a slight modification of the journal article in Advances and Applications in Discrete Mathematics, 13 (1) (2014), 43-6

Harmonic motion and Cassini ovals

We consider a two-dimensional free harmonic oscillator where the initial position is fixed and the initial velocity can change direction. All possible orbits are ellipses and their enveloping curve is an ellipse too. We show that the locus of the foci of all elliptical orbits is a Cassini oval. Depending on the magnitude of the initial velocity we observe all three kinds of Cassini ovals, one of which is the lemniscate of Bernoulli. These Cassini ovals have the same foci as the enveloping ellipse.Comment: Eight figures created with GeoGebra. The paper has been presented in a talk at Ohio Northern University in September 201

Euler Sums of Hyperharmonic Numbers

The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: {\sigma}(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of series of Hurwitz zeta function values. This is a generalization of a result of Mez\H{o} and Dil. We also provide an explicit evaluation of {\sigma}(r,m) in a closed form in terms of zeta values and Stirling numbers of the first kind. Furthermore, we evaluate several other series involving hyperharmonic numbers.Comment: 9 page